How to compare infinity?

Question

Mathematics

Given:
An infinite identical row A and B of something (numbers, boxes, commands...).
Question:
Which is less A or B-1?
PS The
question is Friday and not very serious. I'm a bit confused so I thought I'd ask.

Answer 1

Infinity is always greater than something and minus infinity is always less, it is incorrect to compare two infinities, if infinity were a type (of type true or false for example infinity) then logically infinity = infinity but in fact they may have a different number of values ​​so infinity != infinity is shorter It's like you can't divide by zero.

Answer 2

In the case of an infinite number of boxes, we have a countable set. This means that the boxes can be numbered with natural numbers (for simplicity, let a number of boxes have a beginning, and it goes infinitely to the right |1| |2| |3| |4| ... |n| ...) To number means to build mutually one-to-one correspondence between boxes and numbers (bijection). If it is possible to construct a bijection for two sets, then they are equivalent.
If you remove one of the boxes, then on all the boxes to the right of it, you can replace the number from i to i-1. We get that all the boxes are numbered, and the cardinality of the set without 1 box remains the same.
In this case, usually the cardinality of a countable set is not used as a number. If we add it as a number to the natural series, then the Peano axioms that define this series are violated. No one bothers to change these axioms, but such an extended series does not describe the real world very well, unlike ordinary natural numbers. That's why they don't usually do that.
And yes, there are still infinities "more" than countable ones. It can be proved that the points on the segment [0, 1) cannot be numbered (a separate question is what are the points on the segment, see Chapter 2 ). The cardinality of a set of points on a segment coincides with the cardinality of a set of points on a line, on a plane, or in a cube. This power is called the continuum. Theoretically , one can buildand many high powers.

Answer 3

It depends what you mean by "compare". If you want to know where there are more elements - in A or A-1, while A is really a "row" (infinite sequence), then they are always the same. If you want to know which row contains more elements, then you need to compare the corresponding boxes and see which value is larger - in A or in A-1. Which elements are more, those will win.
For example, if A=1,2,3,4,... then A-1=2,3,4,5,... and A-1 > A in all indices. If A=1/1,1/2,1/3,..., then A-1 < A. If the series is more tricky, for example, 1,-2,3,-4,..., then for to determine which indexes are larger, you need to use such an object as a non-principal ultrafilter - an excellent thing, but, unfortunately, not constructive. The result is an object that is the simplest optionhyperreal numbers , opening the way to non-standard analysis.

Answer 4

Compare powers of infinity. In this case, they are equal.
But the infinity of all even numbers is less than the infinity of all natural numbers.

Answer 5

Function comparison. Infinitely small and infinitely b...